| L(s) = 1 | − 3-s − 0.597·5-s + 2.60·7-s + 9-s − 4.70·11-s − 4.30·13-s + 0.597·15-s + 2.94·17-s − 4.08·19-s − 2.60·21-s + 2.33·23-s − 4.64·25-s − 27-s − 2.00·29-s + 5.83·31-s + 4.70·33-s − 1.55·35-s − 1.82·37-s + 4.30·39-s + 8.73·41-s + 11.8·43-s − 0.597·45-s − 2.91·47-s − 0.188·49-s − 2.94·51-s + 9.63·53-s + 2.80·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.267·5-s + 0.986·7-s + 0.333·9-s − 1.41·11-s − 1.19·13-s + 0.154·15-s + 0.713·17-s − 0.937·19-s − 0.569·21-s + 0.486·23-s − 0.928·25-s − 0.192·27-s − 0.372·29-s + 1.04·31-s + 0.818·33-s − 0.263·35-s − 0.299·37-s + 0.689·39-s + 1.36·41-s + 1.80·43-s − 0.0890·45-s − 0.425·47-s − 0.0268·49-s − 0.411·51-s + 1.32·53-s + 0.378·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.170279346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.170279346\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 0.597T + 5T^{2} \) |
| 7 | \( 1 - 2.60T + 7T^{2} \) |
| 11 | \( 1 + 4.70T + 11T^{2} \) |
| 13 | \( 1 + 4.30T + 13T^{2} \) |
| 17 | \( 1 - 2.94T + 17T^{2} \) |
| 19 | \( 1 + 4.08T + 19T^{2} \) |
| 23 | \( 1 - 2.33T + 23T^{2} \) |
| 29 | \( 1 + 2.00T + 29T^{2} \) |
| 31 | \( 1 - 5.83T + 31T^{2} \) |
| 37 | \( 1 + 1.82T + 37T^{2} \) |
| 41 | \( 1 - 8.73T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 2.91T + 47T^{2} \) |
| 53 | \( 1 - 9.63T + 53T^{2} \) |
| 59 | \( 1 - 6.28T + 59T^{2} \) |
| 61 | \( 1 + 8.58T + 61T^{2} \) |
| 67 | \( 1 + 5.46T + 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 + 2.56T + 73T^{2} \) |
| 79 | \( 1 - 4.32T + 79T^{2} \) |
| 83 | \( 1 + 8.40T + 83T^{2} \) |
| 89 | \( 1 - 8.59T + 89T^{2} \) |
| 97 | \( 1 - 6.10T + 97T^{2} \) |
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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
−8.181146154632335548187107811728, −7.70441129648027196632519082190, −7.21854562807918338063884597772, −6.06521005072070582478985137952, −5.39007231764089158429364841111, −4.77345588218554355945198908277, −4.11455078701161452518728828075, −2.77788761258639635616981621206, −2.02704889271303671908603000836, −0.62120430203534693399459627492,
0.62120430203534693399459627492, 2.02704889271303671908603000836, 2.77788761258639635616981621206, 4.11455078701161452518728828075, 4.77345588218554355945198908277, 5.39007231764089158429364841111, 6.06521005072070582478985137952, 7.21854562807918338063884597772, 7.70441129648027196632519082190, 8.181146154632335548187107811728
