| L(s) = 1 | + 1.49·3-s − 1.09i·5-s − i·7-s − 0.770·9-s − i·11-s + (0.907 + 3.48i)13-s − 1.63i·15-s − 2.55·17-s + 1.17i·19-s − 1.49i·21-s + 3.34·23-s + 3.80·25-s − 5.62·27-s + 3.03·29-s − 9.18i·31-s + ⋯ |
| L(s) = 1 | + 0.862·3-s − 0.489i·5-s − 0.377i·7-s − 0.256·9-s − 0.301i·11-s + (0.251 + 0.967i)13-s − 0.421i·15-s − 0.619·17-s + 0.268i·19-s − 0.325i·21-s + 0.696·23-s + 0.760·25-s − 1.08·27-s + 0.562·29-s − 1.65i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.231925410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.231925410\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (-0.907 - 3.48i)T \) |
| good | 3 | \( 1 - 1.49T + 3T^{2} \) |
| 5 | \( 1 + 1.09iT - 5T^{2} \) |
| 17 | \( 1 + 2.55T + 17T^{2} \) |
| 19 | \( 1 - 1.17iT - 19T^{2} \) |
| 23 | \( 1 - 3.34T + 23T^{2} \) |
| 29 | \( 1 - 3.03T + 29T^{2} \) |
| 31 | \( 1 + 9.18iT - 31T^{2} \) |
| 37 | \( 1 + 6.67iT - 37T^{2} \) |
| 41 | \( 1 + 3.39iT - 41T^{2} \) |
| 43 | \( 1 - 5.02T + 43T^{2} \) |
| 47 | \( 1 + 4.66iT - 47T^{2} \) |
| 53 | \( 1 - 9.15T + 53T^{2} \) |
| 59 | \( 1 - 1.21iT - 59T^{2} \) |
| 61 | \( 1 - 6.33T + 61T^{2} \) |
| 67 | \( 1 + 5.15iT - 67T^{2} \) |
| 71 | \( 1 + 7.39iT - 71T^{2} \) |
| 73 | \( 1 + 13.4iT - 73T^{2} \) |
| 79 | \( 1 + 1.16T + 79T^{2} \) |
| 83 | \( 1 - 2.86iT - 83T^{2} \) |
| 89 | \( 1 + 3.44iT - 89T^{2} \) |
| 97 | \( 1 + 4.42iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.422375989073608927832813472293, −7.68214326394915942351086345573, −6.94215432222485717652565618121, −6.11840263807046411328698597703, −5.26867284931466873310025818583, −4.28990081553110533605431477290, −3.74620953179479373167675341906, −2.69894899409794953781217253079, −1.90914030128754818222551486260, −0.59741644244608566096159912931,
1.19756697992766458960919311181, 2.68956473811193907626058358141, 2.79336051983832464971565294394, 3.81421102830029527377776462096, 4.88791691387295821270665219090, 5.58087516042695947786543837165, 6.59016075103514636621452502672, 7.10732259912318120452901392568, 8.080910687130024807090019852905, 8.574832877027767808723086771089
