| L(s) = 1 | + 1.16·3-s − 3.04·5-s − 0.946·7-s − 1.64·9-s + 5.26·11-s + 4.98·13-s − 3.55·15-s − 4.70·17-s − 6.27·19-s − 1.10·21-s + 4.26·23-s + 4.27·25-s − 5.41·27-s + 9.05·29-s − 6.05·31-s + 6.14·33-s + 2.88·35-s + 0.0493·37-s + 5.80·39-s + 4.78·41-s + 8.15·43-s + 4.99·45-s − 12.9·47-s − 6.10·49-s − 5.48·51-s + 3.38·53-s − 16.0·55-s + ⋯ |
| L(s) = 1 | + 0.673·3-s − 1.36·5-s − 0.357·7-s − 0.547·9-s + 1.58·11-s + 1.38·13-s − 0.916·15-s − 1.14·17-s − 1.44·19-s − 0.240·21-s + 0.889·23-s + 0.854·25-s − 1.04·27-s + 1.68·29-s − 1.08·31-s + 1.06·33-s + 0.487·35-s + 0.00811·37-s + 0.929·39-s + 0.747·41-s + 1.24·43-s + 0.744·45-s − 1.88·47-s − 0.872·49-s − 0.768·51-s + 0.464·53-s − 2.16·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.668016399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.668016399\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 1.16T + 3T^{2} \) |
| 5 | \( 1 + 3.04T + 5T^{2} \) |
| 7 | \( 1 + 0.946T + 7T^{2} \) |
| 11 | \( 1 - 5.26T + 11T^{2} \) |
| 13 | \( 1 - 4.98T + 13T^{2} \) |
| 17 | \( 1 + 4.70T + 17T^{2} \) |
| 19 | \( 1 + 6.27T + 19T^{2} \) |
| 23 | \( 1 - 4.26T + 23T^{2} \) |
| 29 | \( 1 - 9.05T + 29T^{2} \) |
| 31 | \( 1 + 6.05T + 31T^{2} \) |
| 37 | \( 1 - 0.0493T + 37T^{2} \) |
| 41 | \( 1 - 4.78T + 41T^{2} \) |
| 43 | \( 1 - 8.15T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 3.38T + 53T^{2} \) |
| 59 | \( 1 - 6.40T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 6.83T + 67T^{2} \) |
| 71 | \( 1 + 0.707T + 71T^{2} \) |
| 73 | \( 1 - 9.30T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 6.10T + 83T^{2} \) |
| 89 | \( 1 - 7.70T + 89T^{2} \) |
| 97 | \( 1 - 7.42T + 97T^{2} \) |
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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
−8.109095542355619228670990905115, −7.08581277232553010461486764563, −6.50761035038448254076426002938, −6.05793923313845592302147225906, −4.69078454930018095315145007564, −4.06883890785381151428415186126, −3.60727741549094544642991336834, −2.90330168276564429830902341788, −1.79493605391344445523007246174, −0.61584702895197887736096309986,
0.61584702895197887736096309986, 1.79493605391344445523007246174, 2.90330168276564429830902341788, 3.60727741549094544642991336834, 4.06883890785381151428415186126, 4.69078454930018095315145007564, 6.05793923313845592302147225906, 6.50761035038448254076426002938, 7.08581277232553010461486764563, 8.109095542355619228670990905115
