| L(s) = 1 | + 2-s + 0.594·3-s + 4-s + 1.59·5-s + 0.594·6-s + 7-s + 8-s − 2.64·9-s + 1.59·10-s + 5.13·11-s + 0.594·12-s + 2.64·13-s + 14-s + 0.947·15-s + 16-s − 6.54·17-s − 2.64·18-s − 5.73·19-s + 1.59·20-s + 0.594·21-s + 5.13·22-s + 0.594·24-s − 2.45·25-s + 2.64·26-s − 3.35·27-s + 28-s + 7.54·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.343·3-s + 0.5·4-s + 0.712·5-s + 0.242·6-s + 0.377·7-s + 0.353·8-s − 0.882·9-s + 0.504·10-s + 1.54·11-s + 0.171·12-s + 0.734·13-s + 0.267·14-s + 0.244·15-s + 0.250·16-s − 1.58·17-s − 0.623·18-s − 1.31·19-s + 0.356·20-s + 0.129·21-s + 1.09·22-s + 0.121·24-s − 0.491·25-s + 0.519·26-s − 0.645·27-s + 0.188·28-s + 1.40·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.699030467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.699030467\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 0.594T + 3T^{2} \) |
| 5 | \( 1 - 1.59T + 5T^{2} \) |
| 11 | \( 1 - 5.13T + 11T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 + 6.54T + 17T^{2} \) |
| 19 | \( 1 + 5.73T + 19T^{2} \) |
| 29 | \( 1 - 7.54T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 - 6.24T + 37T^{2} \) |
| 41 | \( 1 + 5.83T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 2.08T + 47T^{2} \) |
| 53 | \( 1 - 9.54T + 53T^{2} \) |
| 59 | \( 1 - 8.78T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 1.49T + 67T^{2} \) |
| 71 | \( 1 + 1.67T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 9.42T + 79T^{2} \) |
| 83 | \( 1 - 7.64T + 83T^{2} \) |
| 89 | \( 1 + 1.67T + 89T^{2} \) |
| 97 | \( 1 - 6.80T + 97T^{2} \) |
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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
−8.065241669369775268280267808484, −6.79230523846959254489999490457, −6.43607753074392675326357473389, −5.96374159903741032359091087598, −5.03780839438364401040476495533, −4.15707845701062635699903905324, −3.80396474582986056838115681877, −2.49628095708376626540982444806, −2.16585119081641227477853795034, −0.993044951670342295043036433692,
0.993044951670342295043036433692, 2.16585119081641227477853795034, 2.49628095708376626540982444806, 3.80396474582986056838115681877, 4.15707845701062635699903905324, 5.03780839438364401040476495533, 5.96374159903741032359091087598, 6.43607753074392675326357473389, 6.79230523846959254489999490457, 8.065241669369775268280267808484
