| L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s − 1.72i·7-s − i·8-s − 9-s − 2.44·11-s − i·12-s + 2.96i·13-s + 1.72·14-s + 16-s + 3.34i·17-s − i·18-s − 2.24·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.652i·7-s − 0.353i·8-s − 0.333·9-s − 0.736·11-s − 0.288i·12-s + 0.823i·13-s + 0.461·14-s + 0.250·16-s + 0.811i·17-s − 0.235i·18-s − 0.515·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.131078762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.131078762\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 1.72iT - 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 - 2.96iT - 13T^{2} \) |
| 17 | \( 1 - 3.34iT - 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 + 9.11iT - 23T^{2} \) |
| 29 | \( 1 - 9.37T + 29T^{2} \) |
| 31 | \( 1 + 6.12T + 31T^{2} \) |
| 37 | \( 1 - 3.30iT - 37T^{2} \) |
| 41 | \( 1 + 6.12T + 41T^{2} \) |
| 43 | \( 1 + 7.06iT - 43T^{2} \) |
| 47 | \( 1 + 4.99iT - 47T^{2} \) |
| 53 | \( 1 - 6.85iT - 53T^{2} \) |
| 59 | \( 1 - 0.182T + 59T^{2} \) |
| 61 | \( 1 - 5.15T + 61T^{2} \) |
| 67 | \( 1 + 11.8iT - 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 6.72iT - 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 9.13iT - 83T^{2} \) |
| 89 | \( 1 - 8.11T + 89T^{2} \) |
| 97 | \( 1 + 9.88iT - 97T^{2} \) |
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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
−8.554063161371834734240944927025, −7.84786350644880407904847904940, −6.86836634059622030336035323646, −6.45069596408382836810701004712, −5.50128478313928261973523054374, −4.60583955332503856844161113933, −4.21548271492121137323729201574, −3.19935908553945791850811485243, −2.02001072732378285781746607946, −0.40014092310692322898056237449,
0.939030286286236118832940552370, 2.10001416485534769283867797372, 2.83550898755175999290049463607, 3.56980585380704965742684179168, 4.86955914082348140737874757198, 5.41416772219670369278990237697, 6.14863687336376811015634059094, 7.22615394194592228407309673330, 7.82975095107296248802284463873, 8.553700774453789794805444841628
