| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 0.533·7-s − 8-s + 9-s − 1.43·11-s − 12-s − 6.57·13-s − 0.533·14-s + 16-s + 0.958·17-s − 18-s + 0.212·19-s − 0.533·21-s + 1.43·22-s − 3.76·23-s + 24-s + 6.57·26-s − 27-s + 0.533·28-s + 6.19·29-s − 2.33·31-s − 32-s + 1.43·33-s − 0.958·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.201·7-s − 0.353·8-s + 0.333·9-s − 0.432·11-s − 0.288·12-s − 1.82·13-s − 0.142·14-s + 0.250·16-s + 0.232·17-s − 0.235·18-s + 0.0488·19-s − 0.116·21-s + 0.305·22-s − 0.784·23-s + 0.204·24-s + 1.28·26-s − 0.192·27-s + 0.100·28-s + 1.15·29-s − 0.419·31-s − 0.176·32-s + 0.249·33-s − 0.164·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7072365319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7072365319\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 0.533T + 7T^{2} \) |
| 11 | \( 1 + 1.43T + 11T^{2} \) |
| 13 | \( 1 + 6.57T + 13T^{2} \) |
| 17 | \( 1 - 0.958T + 17T^{2} \) |
| 19 | \( 1 - 0.212T + 19T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 - 6.19T + 29T^{2} \) |
| 31 | \( 1 + 2.33T + 31T^{2} \) |
| 37 | \( 1 + 4.06T + 37T^{2} \) |
| 41 | \( 1 + 7.94T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 3.23T + 53T^{2} \) |
| 59 | \( 1 - 7.52T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 6.91T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 - 5.62T + 89T^{2} \) |
| 97 | \( 1 - 5.56T + 97T^{2} \) |
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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
−8.296173324698738606161483462435, −7.912237141092534853157528685203, −7.00187378890523813363019578203, −6.56566890559305153731644231345, −5.35631047609190031709798347069, −5.03796659456933699751933378603, −3.90091949023107411386942878034, −2.70090730864254303154845837031, −1.91173725695410114690232985009, −0.54071937685458495413681151446,
0.54071937685458495413681151446, 1.91173725695410114690232985009, 2.70090730864254303154845837031, 3.90091949023107411386942878034, 5.03796659456933699751933378603, 5.35631047609190031709798347069, 6.56566890559305153731644231345, 7.00187378890523813363019578203, 7.912237141092534853157528685203, 8.296173324698738606161483462435
