| L(s) = 1 | + 4·2-s + 12·4-s + 9·5-s + 19·7-s + 32·8-s + 36·10-s + 24·11-s + 61·13-s + 76·14-s + 80·16-s − 3·17-s + 133·19-s + 108·20-s + 96·22-s − 69·23-s + 47·25-s + 244·26-s + 228·28-s − 237·29-s + 211·31-s + 192·32-s − 12·34-s + 171·35-s + 262·37-s + 532·38-s + 288·40-s − 468·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.804·5-s + 1.02·7-s + 1.41·8-s + 1.13·10-s + 0.657·11-s + 1.30·13-s + 1.45·14-s + 5/4·16-s − 0.0428·17-s + 1.60·19-s + 1.20·20-s + 0.930·22-s − 0.625·23-s + 0.375·25-s + 1.84·26-s + 1.53·28-s − 1.51·29-s + 1.22·31-s + 1.06·32-s − 0.0605·34-s + 0.825·35-s + 1.16·37-s + 2.27·38-s + 1.13·40-s − 1.78·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.766572861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.766572861\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 9 T + 34 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 19 T + 540 T^{2} - 19 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 24 T + 1861 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 61 T + 5088 T^{2} - 61 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 9592 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 7 p T + 12234 T^{2} - 7 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 p T + 25288 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 237 T + 51244 T^{2} + 237 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 211 T + 68586 T^{2} - 211 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 262 T + 72162 T^{2} - 262 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 468 T + 191653 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 p T + 24783 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 483 T + 212812 T^{2} + 483 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 150 T + 257074 T^{2} + 150 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 168 T + 416869 T^{2} + 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1049 T + 675906 T^{2} + 1049 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1166 T + 907395 T^{2} + 1166 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 312 T + 498238 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 311 T + 698028 T^{2} + 311 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 349 T + 976602 T^{2} - 349 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1221 T + 1412098 T^{2} + 1221 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 492 T + 1092454 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 128 T + 1715097 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.86193395405478908583846154276, −11.99014623246282991841007437024, −11.74079511605902335600045930587, −11.42072697883217814226210986080, −10.84422874897085128972421068478, −10.33610202205239892002473890421, −9.604497963808181260243034377935, −9.252689686930681629740162265249, −8.295352048796842345401238760521, −7.976166263009545346813965462813, −7.24623132831468950604676318424, −6.56542404486668705066529229283, −5.97557494236137692992582273031, −5.69740618513563164423064690680, −4.83020451228871719287706149927, −4.49360555502426044675850688433, −3.50611129512452170337854355701, −3.04537032240315496925935053444, −1.70914247085788237108016117633, −1.43781608275696981042008320935,
1.43781608275696981042008320935, 1.70914247085788237108016117633, 3.04537032240315496925935053444, 3.50611129512452170337854355701, 4.49360555502426044675850688433, 4.83020451228871719287706149927, 5.69740618513563164423064690680, 5.97557494236137692992582273031, 6.56542404486668705066529229283, 7.24623132831468950604676318424, 7.976166263009545346813965462813, 8.295352048796842345401238760521, 9.252689686930681629740162265249, 9.604497963808181260243034377935, 10.33610202205239892002473890421, 10.84422874897085128972421068478, 11.42072697883217814226210986080, 11.74079511605902335600045930587, 11.99014623246282991841007437024, 12.86193395405478908583846154276
