| L(s) = 1 | + 3-s + 5-s − 2.56·7-s + 9-s − 2.56·11-s + 2·13-s + 15-s − 3.12·17-s + 7.68·19-s − 2.56·21-s − 1.43·23-s + 25-s + 27-s + 7.12·29-s − 31-s − 2.56·33-s − 2.56·35-s − 3.12·37-s + 2·39-s + 7.12·41-s − 12.8·43-s + 45-s − 5.12·47-s − 0.438·49-s − 3.12·51-s + 7.43·53-s − 2.56·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.968·7-s + 0.333·9-s − 0.772·11-s + 0.554·13-s + 0.258·15-s − 0.757·17-s + 1.76·19-s − 0.558·21-s − 0.299·23-s + 0.200·25-s + 0.192·27-s + 1.32·29-s − 0.179·31-s − 0.445·33-s − 0.432·35-s − 0.513·37-s + 0.320·39-s + 1.11·41-s − 1.95·43-s + 0.149·45-s − 0.747·47-s − 0.0626·49-s − 0.437·51-s + 1.02·53-s − 0.345·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.371175850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.371175850\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 - 7.68T + 19T^{2} \) |
| 23 | \( 1 + 1.43T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 37 | \( 1 + 3.12T + 37T^{2} \) |
| 41 | \( 1 - 7.12T + 41T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 47 | \( 1 + 5.12T + 47T^{2} \) |
| 53 | \( 1 - 7.43T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 - 7.68T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 4.31T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.102653435309411881512184918512, −6.91842935713703258578292829214, −6.81661461447846374973625135662, −5.72538473438087726042693027649, −5.20995794693537539466641240697, −4.22106588411592074670113032366, −3.31749572217411944440369375061, −2.84662738754063065505000345860, −1.92370384464123265147536759049, −0.74490729290079258126477942500,
0.74490729290079258126477942500, 1.92370384464123265147536759049, 2.84662738754063065505000345860, 3.31749572217411944440369375061, 4.22106588411592074670113032366, 5.20995794693537539466641240697, 5.72538473438087726042693027649, 6.81661461447846374973625135662, 6.91842935713703258578292829214, 8.102653435309411881512184918512
